If a medial area is subtracted from a rational area, then the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line.

Let the medial area *BD* be subtracted from the rational area *BC*.

I say that the side of the remainder *EC* becomes one of two irrational straight lines, either an apotome or a minor straight line.

Set out a rational straight line *FG*, to *FG* apply the rectangular parallelogram *GH* equal to *BC*, and subtract *GK* equal to *DB*. Then the remainder *EC* equals *LH*.

Since, then, *BC* is rational, and *BD* medial, while *BC* equals *GH*, and *BD* equals *GK*, therefore *GH* is rational, and *GK* is medial.

And they are applied to the rational straight line *FG*, therefore *FH* is rational and commensurable in length with *FG*, while *FK* is rational and incommensurable in length with *FG*. Therefore *FH* is incommensurable in length with *FK*.

Therefore *FH* and *FK* are rational straight lines commensurable in square only. Therefore *KH* is an apotome, and *KF* the annex to it.

Now the square on *HF* is greater than the square on *FK* by the square on a straight line either commensurable with *HF* or not commensurable.

First, let the square on it be greater by the square on a straight line commensurable with it.

Now the whole *HF* is commensurable in length with the rational straight line *FG* set out, therefore *KH* is a first apotome.

But the side of the rectangle contained by a rational straight line and a first apotome is an apotome. Therefore the side of *LH*, that is, of *EC*, is an apotome.

But, if the square on *HF* is greater than the square on *FK* by the square on a straight line incommensurable with *HF*, while the whole *FH* is commensurable in length with the rational straight line *FG* set out, then *KH* is a fourth apotome.

But the side of the rectangle contained by a rational straight line and a fourth apotome is minor.

Therefore, *if a medial area is subtracted from a rational area, then the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line.*

Q.E.D.